The /gnomon/ is a scientific instrument that can be used for finding the
declination of the sun through the year, among other things. It is one
of the first scientific instruments ever made, originating with the
Chaldean astronomers of Babylon and from there brought to the Greek
world. It is still very instructive, an excellent example of random and
systematic errors of measurement, vividly demonstrating astronomical
principles. The Greek gnwmwn, originally meaning /one who knows/, or
/judge/, was also used for the /index/ of a sundial, as well as for the
carpenter's square. It is easy to make a good portable gnomon, and easy
to use it.
To make a gnomon, you will need a straight piece of wood for the base (I
used a redwood 1 x 3 by 22 in. long), a length of metal rod (mine was
1/8 in. diameter brass rod about 10 in. long), a piece of millimeter
paper, a small spirit level, and some household cement. The rod should
be straight. Finish the wood as nicely as you want, then drill a 1/8 in.
hole about 1 in. from one end on the center line. You must use a drill
press so that the hole is perpendicular, to hold the gnomon at right
angles to the base. Great accuracy in perpendicularity is not required.
Saw off the piece of rod with a hacksaw, and finish the ends with a
file. Then cement the rod in the hole, getting it as closely as possible
to a right angle from the board. Prepare a strip of millimeter paper
about 4 cm wide long enough to go from a couple of inches from the rod
to near the end of the board. This may have to be in two pieces. With a
pencil and ruler, draw a line down the middle of the strip. Cement the
strip to the board so that the extended line passes through the base of
the rod, and that some centimeter line is exactly a certain number of
centimeters from the center of the rod. When this is done, mark every 50
or 100 mm on the line to make readings easy. Make sure that the base and
the scale are long enough to measure the lowest elevation of the sun at
noon (about 66° minus your latitude). Measure the height of the top of
the rod from the board, and note the value in mm on the paper strip. The
rod is the actual gnomon, while the base used for making a measurement.
A small pocket spirit level can be obtained at low cost, and should be
cemented to the board alongside the scale. All this can be done in
fifteen or twenty minutes!
To use the gnomon, set it in the sun and rotate the base so that the
shadow of the rod falls on the center line. The scale for reading the
length of the shadow must be accurately horizontal, so use the spirit
level to make sure of this, shimming the base as necessary with pieces
of cardboard. Now, read the length of the shadow on the scale to the
nearest millimeter. Note that the end of the shadow is not sharp - why
not? - so measure to the center of the penumbra. The altitude of the sun
is the arctangent of the ratio of the length of the gnomon to the length
of the shadow, easily computed on a scientific pocket calculator. My
gnomon can measure altitudes from 28° to 78° , or just about enough to
measure the noon elevation of the sun for any date at my latitude of 40°
. I should have made the board a little longer, since the minimum solar
altitude at noon is 26° . If the board is set accurately north-south on
the meridian, local noon occurs when the shadow is on the center line.
This is the time to measure the altitude of the sun to determine its
declination. When daylight savings time is in effect, noon is really at
1 pm. The sun is not exactly on the meridian at standard time noon, but
the difference is not great.
The declination of the sun d (its angle north or south of the celestial
equator) is the sum of the solar altitude a at noon and your latitude f
, less 90° , as you can easily deduce from the Figure. Z is the zenith,
P the pole, E the equator, and N,S are the north and south points. On a
piece of graph paper, plot the declination along the y-axis, and the day
of the year along the x-axis. Number the days beginning with March 21,
which is near the vernal equinox, and allow for declinations in the
range ± 30° . Plot the declination for every day that is sunny at noon -
in Denver this can be done with few gaps - and missing a few days will
not matter in any case. After you have done this for a full year, you
will appreciate the sun's motion much more clearly.
Today, 5 September 1998, at apparent noon, the length of the shadow was
158 mm, giving an altitude of 56.7° , and a solar declination of 56.7° +
39.7° - 90° = +6.6° . The Astronomical Almanac 1998, page C14, gives the
declination of the sun at this time as +6.7° , so my result is very
good. I am pretty sure I can read the shadow to about 1 mm more or less.
This is an estimate of a /random/ error of measurement. Taking shadow
lengths of 157 and 159 mm, I find that the corresponding difference in
elevation is only 0.4° , so the accuracy of the result is consistent
with this. If you are fortunate, your first attempt will be a degree or
so off, and you can have the satisfaction of tracking down the reason
for the discrepancy. I had to do this before I obtained a result free of
systematic error. Don't give up until you can get results accurate to
within half a degree, which this instrument is capable of.
A /systematic/ error is a bias that is not a result of the ordinary
uncertainty in reading a scale, or of other random fluctuations. It
cannot be removed by averaging repeated measurements. The systematic
errors of the gnomon are mainly constant ones. For example, an
inaccurate level, wrong height of the gnomon, and incorrect latitude all
will cause the measured value to deviate from the correct value. If you
place the gnomon in the same place for all measurements, without
leveling, there will be a constant systematic error in elevation. Each
measurement can be corrected for this error once it has been determined,
saving the bother of leveling each time. The level and latitude should
be known to 0.1° (1/32" in 18" for level, and within 6 miles for
latitude). Your latitude can be obtained from a USGS topographic map.
[Sundial] If the gnomon points to the celestial pole instead of being
vertical, the shadow moves uniformly with the hour angle of the sun, as
in the /equatorial dial/ shown in the picture, which is at Cranmer Park
in Denver, Colorado. In the winter, the lower face of the dial must be
used instead. The dial shows apparent solar time, of course, and must be
adjusted by the equation of time and for the longitude to show standard
time.
Use the gnomon to measure the height of a tree. This can be done at any
time, not just at noon, and does not require any trigonometry. If you
know the declination of the sun at the particular time, the gnomon can
be used to determine the latitude. This information is given very
accurately in the Astronomical Almanac for the year, but an approximate
value can be determined from the date in any year, for example from your
annual plot. The gnomon can also be used to determine the time, when you
know the sun's declination and the meridian, and the direction of the
meridian, when you know the time. This does require some trigonometry,
spherical trigonometry in fact. The gnomon can easily be used to
determine the approximate date - figure out how to do this. This
operation has been carried out to determine the date of the vernal
equinox since very ancient times. What are some other ways of finding
the vernal equinox? The full moon, as well as the sun, casts a
measurable shadow. Mark the end of the shadow in the dark, then read the
length of the shadow with a light.
I used my gnomon to measure and plot the declination of the sun
throughout a year <http://www.du.edu/%7Ejcalvert/astro/gnoplot.htm>.
This gave a vivid appreciation for the movement of the sun. It was very
pleasing to see the plot develop, and showed how to find the latitude to
a degree or better with such a simple instrument. With a permanent
gnomon, properly fixed, the instant of apparent noon could be obtained
and compared with the standard time, thereby finding the equation of
time and the variations in the sun's motion through the year.